The Eckmann-Hilton argument and higher operads

M. A. Batanin*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    23 Citations (Scopus)

    Abstract

    The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is 'the same' as a braided monoidal category. In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, ..., one (n - 1)-arrow algebras of A is isomorphic to the category of algebras of Symn (A). Under some mild conditions, we present an explicit formula for Symn (A) which involves taking the colimit over a remarkable categorical symmetric operad. We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.

    Original languageEnglish
    Pages (from-to)334-385
    Number of pages52
    JournalAdvances in Mathematics
    Volume217
    Issue number1
    DOIs
    Publication statusPublished - 15 Jan 2008

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